完美匹配中的边权对称性层次结构
A hierarchy of edge-weight symmetries in perfect matchings
AI总结：
本文研究了图中边权对称性的结构性质，定义并分析了边权属性的层次结构，证明了在双色图中这些属性等价，而在一般图中提供了紧致的结构刻画，并通过反例证明了单个边不足以消除所有极值完美匹配。
摘要：
受精确权重完美匹配问题和寻找第ℓ-小完美匹配的参数化算法启发，我们研究了图中边权对称性的结构性质。El Maalouly等人（ESA 2025）证明了排除所有权重不超过图中第(ℓ-1)小可能值的完美匹配需要固定至多2(ℓ-1)条边（在非双色图中）或至多ℓ-1条边（在双色图中）。一个自然的开放问题是，是否固定单个边就足以在全局最小和最大权重不同的情况下改变完美匹配的极值权重。为此，我们定义并分析了逐渐弱化的边权属性层次：节点诱导权重、偶走和环对称性、完美匹配相等性和边min-max性质。我们推导了这些条件的基本层次结构，并证明在双色图中它们等价。对于一般图，我们基于块和紧致割分解提供了紧致的结构刻画，其中偶环对称性和完美匹配相等性迫使节点诱导权重。最后，我们通过构造一个满足边min-max性质但违反完美匹配相等性的匹配覆盖非双色图，证明了单个边不总是足以消除所有极值完美匹配，从而证明了ℓ=2时2(ℓ-1)的紧致性。我们还讨论了该框架向b-因子和树状结构的扩展。
英文摘要：
Motivated by the exact weight perfect matching problem and recent parameterized algorithms for finding an $\ell$-th smallest perfect matching, we study structural properties of edge-weight symmetries in graphs. Recent work by El Maalouly et al. (ESA 2025) showed that excluding all perfect matchings whose weight is at most the $(\ell - 1)$-th smallest possible value in the graph requires fixing at most $2(\ell-1)$ edges in non-bipartite graphs and at most $\ell-1$ edges in bipartite graphs. A natural open question is whether fixing a single edge is always sufficient to shift the extreme (minimum or maximum) weight of a perfect matching when the global minimum and maximum weights differ. To address this, we define and analyze a hierarchy of progressively weaker edge-weight properties: node-induced weights, even walk and cycle symmetries, perfect matching equality, and the edge min-max property. We derive a basic hierarchy among these conditions and show that they become equivalent in bipartite graphs. For general graphs, we provide tight structural characterizations, based on block and tight cut decompositions, under which even cycle symmetry and perfect matching equality force node-induced weights. Finally, we resolve the motivating open question in the negative by constructing a matching-covered non-bipartite graph that satisfies the edge min-max property (every edge is contained in a minimum-weight perfect matching and a maximum-weight one) but violates perfect matching equality (all perfect matchings have the same weight). This counterexample shows that a single edge is not always sufficient to eliminate all minimum-weight or maximum-weight perfect matchings, thereby proving the tightness of the $2(\ell-1)$ bound for $\ell=2$. We also discuss extensions of this framework to $b$-factors and arborescences.
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